The Linear Combination Method Is Applied To A System Of Equations As Shown:${ \begin{array}{r} 4(0.25x + 0.5y = 3.75) \rightarrow X + 2y = 15 \ \frac{1}{4}(4x - 8y = 12) \rightarrow X - 2y = 3 \ 2x = 18 \end{array} }$What Is The Solution

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Introduction

In mathematics, a system of equations is a set of equations that involve multiple variables. Solving a system of equations can be a challenging task, but the linear combination method provides a powerful tool for finding the solution. This method involves combining two or more equations to eliminate one or more variables, making it easier to solve for the remaining variables. In this article, we will explore the linear combination method and apply it to a system of equations to find the solution.

What is the Linear Combination Method?

The linear combination method is a technique used to solve systems of equations by combining two or more equations to eliminate one or more variables. This method involves multiplying one or more equations by a constant and then adding or subtracting the resulting equations to eliminate a variable. The resulting equation is then solved for the remaining variables.

How to Apply the Linear Combination Method

To apply the linear combination method, follow these steps:

  1. Identify the equations: Identify the system of equations that you want to solve.
  2. Determine the variables: Determine the variables that you want to solve for.
  3. Choose the equations: Choose two or more equations that involve the variables you want to solve for.
  4. Multiply the equations: Multiply one or more equations by a constant to make the coefficients of one or more variables the same.
  5. Add or subtract the equations: Add or subtract the resulting equations to eliminate a variable.
  6. Solve for the remaining variables: Solve the resulting equation for the remaining variables.

Applying the Linear Combination Method to a System of Equations

Let's apply the linear combination method to the system of equations shown below:

4(0.25x+0.5y=3.75)→x+2y=1514(4x−8y=12)→x−2y=32x=18{ \begin{array}{r} 4(0.25x + 0.5y = 3.75) \rightarrow x + 2y = 15 \\ \frac{1}{4}(4x - 8y = 12) \rightarrow x - 2y = 3 \\ 2x = 18 \end{array} }

To apply the linear combination method, we will first multiply the first equation by 4 and the second equation by 1. This will make the coefficients of x the same.

4(x+2y=15)→4x+8y=60(x−2y=3){ \begin{array}{r} 4(x + 2y = 15) \rightarrow 4x + 8y = 60 \\ (x - 2y = 3) \end{array} }

Next, we will add the two equations to eliminate the variable y.

(4x+8y=60)+(x−2y=3)→5x+6y=63{ \begin{array}{r} (4x + 8y = 60) + (x - 2y = 3) \rightarrow 5x + 6y = 63 \end{array} }

Now, we will multiply the third equation by 5 to make the coefficients of x the same.

5(2x=18)→10x=90{ \begin{array}{r} 5(2x = 18) \rightarrow 10x = 90 \end{array} }

Next, we will subtract the two equations to eliminate the variable x.

(5x+6y=63)−(10x=90)→−4x+6y=−27{ \begin{array}{r} (5x + 6y = 63) - (10x = 90) \rightarrow -4x + 6y = -27 \end{array} }

Now, we have two equations with two variables. We can solve for the variables x and y by using the substitution method.

Solving for the Variables x and y

To solve for the variables x and y, we will use the substitution method. We will substitute the value of x from the third equation into the first equation.

5x+6y=6310x=90{ \begin{array}{r} 5x + 6y = 63 \\ 10x = 90 \end{array} }

Substituting the value of x into the first equation, we get:

5(9)+6y=6345+6y=63{ \begin{array}{r} 5(9) + 6y = 63 \\ 45 + 6y = 63 \end{array} }

Subtracting 45 from both sides, we get:

6y=18{ \begin{array}{r} 6y = 18 \end{array} }

Dividing both sides by 6, we get:

y=3{ \begin{array}{r} y = 3 \end{array} }

Now that we have the value of y, we can substitute it into the third equation to find the value of x.

2x=18y=3{ \begin{array}{r} 2x = 18 \\ y = 3 \end{array} }

Substituting the value of y into the third equation, we get:

2x=182x=18{ \begin{array}{r} 2x = 18 \\ 2x = 18 \end{array} }

Dividing both sides by 2, we get:

x=9{ \begin{array}{r} x = 9 \end{array} }

Therefore, the solution to the system of equations is x = 9 and y = 3.

Conclusion

In conclusion, the linear combination method is a powerful tool for solving systems of equations. By combining two or more equations to eliminate one or more variables, we can make it easier to solve for the remaining variables. In this article, we applied the linear combination method to a system of equations and found the solution. We hope that this article has provided a clear understanding of the linear combination method and how to apply it to solve systems of equations.

References

Frequently Asked Questions

  • What is the linear combination method?
    • The linear combination method is a technique used to solve systems of equations by combining two or more equations to eliminate one or more variables.
  • How do I apply the linear combination method?
    • To apply the linear combination method, follow these steps: identify the equations, determine the variables, choose the equations, multiply the equations, add or subtract the equations, and solve for the remaining variables.
  • What are the advantages of the linear combination method?
    • The linear combination method is a powerful tool for solving systems of equations because it allows us to eliminate one or more variables, making it easier to solve for the remaining variables.
      Frequently Asked Questions: The Linear Combination Method ===========================================================

Q: What is the linear combination method?

A: The linear combination method is a technique used to solve systems of equations by combining two or more equations to eliminate one or more variables.

Q: How do I apply the linear combination method?

A: To apply the linear combination method, follow these steps:

  1. Identify the equations: Identify the system of equations that you want to solve.
  2. Determine the variables: Determine the variables that you want to solve for.
  3. Choose the equations: Choose two or more equations that involve the variables you want to solve for.
  4. Multiply the equations: Multiply one or more equations by a constant to make the coefficients of one or more variables the same.
  5. Add or subtract the equations: Add or subtract the resulting equations to eliminate a variable.
  6. Solve for the remaining variables: Solve the resulting equation for the remaining variables.

Q: What are the advantages of the linear combination method?

A: The linear combination method is a powerful tool for solving systems of equations because it allows us to eliminate one or more variables, making it easier to solve for the remaining variables.

Q: When should I use the linear combination method?

A: You should use the linear combination method when you have a system of equations with multiple variables and you want to eliminate one or more variables to make it easier to solve for the remaining variables.

Q: Can I use the linear combination method with any type of equation?

A: Yes, you can use the linear combination method with any type of equation, including linear equations, quadratic equations, and polynomial equations.

Q: How do I know if the linear combination method will work for my system of equations?

A: To determine if the linear combination method will work for your system of equations, you need to check if the equations are linearly independent. If the equations are linearly independent, then the linear combination method will work.

Q: What are some common mistakes to avoid when using the linear combination method?

A: Some common mistakes to avoid when using the linear combination method include:

  • Not checking if the equations are linearly independent: If the equations are not linearly independent, then the linear combination method will not work.
  • Not multiplying the equations by the correct constant: If you multiply the equations by the wrong constant, then you may not eliminate the variable correctly.
  • Not adding or subtracting the equations correctly: If you add or subtract the equations incorrectly, then you may not eliminate the variable correctly.

Q: Can I use the linear combination method to solve systems of equations with more than two variables?

A: Yes, you can use the linear combination method to solve systems of equations with more than two variables. However, you will need to use the method multiple times to eliminate each variable.

Q: How do I know if the linear combination method is the best method to use for my system of equations?

A: To determine if the linear combination method is the best method to use for your system of equations, you need to consider the following factors:

  • The number of variables: If you have a system of equations with multiple variables, then the linear combination method may be a good choice.
  • The type of equations: If you have a system of equations with linear equations, then the linear combination method may be a good choice.
  • The complexity of the equations: If you have a system of equations with complex equations, then the linear combination method may not be the best choice.

Conclusion

In conclusion, the linear combination method is a powerful tool for solving systems of equations. By combining two or more equations to eliminate one or more variables, we can make it easier to solve for the remaining variables. In this article, we answered some frequently asked questions about the linear combination method and provided some tips and tricks for using the method effectively.

References

Additional Resources

  • Linear Combination Method Tutorial: This tutorial provides a step-by-step guide to using the linear combination method to solve systems of equations.
  • Solving Systems of Equations with the Linear Combination Method: This video provides a visual explanation of how to use the linear combination method to solve systems of equations.
  • Linear Combination Method Practice Problems: This worksheet provides practice problems for using the linear combination method to solve systems of equations.